Quadratics over the complex numbers

Solve quadratic equations over the complex numbers and connect their roots to factorization. This practice set uses completing the square, the quadratic formula, complex arithmetic and direct substitution. It includes monic and nonmonic equations, exact irrational imaginary parts and factorizations that retain the original leading coefficient.

Assumptions are central to classifying roots. For real coefficients and a nonzero quadratic coefficient, the discriminant distinguishes two real roots, a repeated real root and a nonreal conjugate pair. Questions also examine a quadratic with complex coefficients, where a nonreal repeated root need not bring its conjugate with it. A parameter case checks what changes when the quadratic coefficient vanishes and the equation becomes linear.

Prerequisites are quadratic equations, the imaginary unit, conjugates and standard complex arithmetic. Both single-select and multiselect tasks progress from classification to exact solutions, reconstruction and justification. Worked explanations show how to simplify negative discriminants, preserve the denominator of the quadratic formula and check a candidate in the original polynomial. The final reasoning task characterizes nonzero purely imaginary roots without using polar form or advanced polynomial theory. Count distinct solutions carefully: a repeated root is one complex number even when it is counted twice by multiplicity.